Michael N. Fried: Didactics and History of Mathematics: Knowledge and Self-Knowledge, Educational studies in mathematics (2007) 66: 202-223.

How to include history of mathematics in mathematics education in a way that is true to the history? This is an old question in the HPM community, and it is not resolved.

In this article, the author looks to Saussure and the theory of semiotics to argue that history of mathematics has an essential role to play in mathematics education. "The historian's and the working mathematician's ways of knowing are complementary."

First, he discusses what is the problem: "If, in the working mathematician's view, the otherness of a historical text is something illusory or merely superficial, the historical point of view is precisely the opposite. From tihs latter pole, a mathematical text is a cultural product, the product of a particular human being or group of human beings living in a particular time."

He gives a short (and welcome) introduction to Saussure, and describes two ways of knowing: synchronic and diachronic. Seen at one single point of time (synchronic), a language (or mathematics) seems static, and you can not see the social forces at play. When you look at a period of time (diachronic), you see how language (or mathematics) is evolving and ever-changing.

The author then gives an example: How Apollonius of Perga's Conics has been interpreted in different ways. Heath wrote that "[Apollonius'] method does not essentially differ from that of modern analytical geometry except that in Apollonius geometrical operations take the place of algebraic calculations" - an interpretation based squarely in the modern mathematics. The author of the article shows how Apollonius could rather give us an alternative way of viewing conics, which makes us more aware of our own modern view.

So what about mathematics education. The author argues that the object of mathematics education should not be only to learn mathematical methods, but "our self-knowledge as mathematical beings". As such, the history is essential to give another perspective and let us learn more about our own views.

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